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A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.

%I A249457 #40 Feb 16 2025 08:33:24
%S A249457 10,100,2890,96100,3237610,109202500,3683712490,124263300100,
%T A249457 4191798484810,141402777864100,4769968258260490,160906295771812900,
%U A249457 5427884341892493610,183099910962324064900,6176546013641762558890,208354665265158340802500,7028469704892605715408010
%N A249457 The numerator of curvatures of touching circles inscribed in a special way in the larger segment of a unit circle divided by a chord of length sqrt(84)/5.
%C A249457 The denominators are conjectured to be A005032.
%C A249457 Refer to comments and links of A240926. Consider a unit circle with a chord of length sqrt(84)/5. This has been chosen such that the larger sagitta has length 7/5. The input, besides the unit circle C, is the circle C_0 with radius R_0 = 7/10, touching the chord and circle C. The following sequence of circles C_n with radii R_n, n >= 1, is obtained from the conditions that C_n touches (i) the circle C, (ii) the chord and (iii) the circle C_(n-1). The curvature of the n-th circle is C_n = 1/R_n, n >= 0, and its numerator is conjectured to be a(n).
%C A249457 If one considers the curvature of touching circles inscribed in the smaller segment (sagitta length 3/5), the rational sequence would be A249458/A169634. See an illustration given in the link.
%C A249457 For the proof and the formula for the rational curvatures of the circles in the larger segment see a comment under A249862. C_n = (5/7)*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). - _Wolfdieter Lang_, Nov 07 2014
%H A249457 Kival Ngaokrajang, <a href="/A249457/a249457.pdf">Illustration of initial terms</a>.
%H A249457 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Sagitta.html">Sagitta</a>.
%H A249457 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials</a>.
%H A249457 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (37,-111,27).
%F A249457 Empirical g.f.: -10*(30*x^2-27*x+1) /((3*x - 1)*(9*x^2-34*x+1)). - _Colin Barker_, Oct 29 2014
%F A249457 From _Wolfdieter Lang_, Nov 07 2014: (Start)
%F A249457 a(n) = 5*(A249862(n) + 3^n) = 5*3^n*(S(n, 34/3) - (17/3)*S(n-1, 34/3) + 1), n >= 0, with Chebyshev's S polynomials (A049310). See the comments on A249862 for the proof.
%F A249457 O.g.f.: 5*((1 - 17*x)/(1 - 34*x + 9*x^2) + 1/(1-3*x)) = 10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)) proving the conjecture of Colin Barker above. (End)
%F A249457 E.g.f.: 5*exp(3*x)*(1 + exp(14*x)*cosh(2*sqrt(70)*x)). - _Stefano Spezia_, Mar 24 2023
%t A249457 LinearRecurrence[{37, -111, 27},{10, 100, 2890},16] (* _Ray Chandler_, Aug 11 2015 *)
%t A249457 CoefficientList[Series[10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 20 2017 *)
%o A249457 (PARI)
%o A249457 {
%o A249457 r=0.7;dn=7;print1(round(dn/r),", ");r1=r;
%o A249457 for (n=1,40,
%o A249457      if (n<=1,ab=2-r,ab=sqrt(ac^2+r^2));
%o A249457      ac=sqrt(ab^2-r^2);
%o A249457      if (n<=1,z=0,z=(Pi/2)-atan(ac/r)+asin((r1-r)/(r1+r));r1=r);
%o A249457      b=acos(r/ab)-z;
%o A249457      r=r*(1-cos(b))/(1+cos(b)); dn=dn*3;
%o A249457      print1(round(dn/r),", ");
%o A249457 )
%o A249457 }
%o A249457 (PARI) x='x+O('x^30); Vec(10*(1 - 27*x + 30*x^2)/((1 - 34*x + 9*x^2)*(1 - 3*x))) \\ _G. C. Greubel_, Dec 20 2017
%o A249457 (Magma) I:=[10,100,2890]; [n le 3 select I[n] else 37*Self(n-1) - 111*Self(n-2) + 27*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Dec 20 2017
%Y A249457 Cf. A005032, A049310, A078986, A097315, A169364, A240926, A247335, A247512, A248834, A249458, A249862.
%K A249457 nonn,frac,easy
%O A249457 0,1
%A A249457 _Kival Ngaokrajang_, Oct 29 2014
%E A249457 Edited. Name and comment small changes, keyword easy added. - _Wolfdieter Lang_, Nov 07 2014
%E A249457 a(16) from _Stefano Spezia_, Mar 24 2023